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The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer. These series consist of the and so on. The following diagrams show a few examples of centered polygonal numbers and their geometric construction. (Compare these diagrams with the diagrams in Polygonal number.)

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  • Centered polygonal number
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  • The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer. These series consist of the and so on. The following diagrams show a few examples of centered polygonal numbers and their geometric construction. (Compare these diagrams with the diagrams in Polygonal number.)
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  • CenteredPolygonalNumber
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  • Centered polygonal number
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  • The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer. These series consist of the * centered triangular numbers 1,4,10,19,31,... (sequence A005448 in OEIS) * centered square numbers 1,5,13,25,41,... (File:OEISicon light.svg A001844) * centered pentagonal numbers 1,6,16,31,51,... (File:OEISicon light.svg A005891) * centered hexagonal numbers 1,7,19,37,61,... (File:OEISicon light.svg A003215) * centered heptagonal numbers 1,8,22,43,71,... (File:OEISicon light.svg A069099) * centered octagonal numbers 1,9,25,49,81,... (File:OEISicon light.svg A016754) * centered nonagonal numbers 1,10,28,55,91,... (File:OEISicon light.svg A060544) * centered decagonal numbers 1,11,31,61,101,... (File:OEISicon light.svg A062786) and so on. The following diagrams show a few examples of centered polygonal numbers and their geometric construction. (Compare these diagrams with the diagrams in Polygonal number.) Centered square numbers Centered hexagonal numbers As can be seen in the above diagrams, the nth centered k-gonal number can be obtained by placing k copies of the (n−1)th triangular number around a central point; therefore, the nth centered k-gonal number can be mathematically represented by Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc. Whereas a prime number p cannot be a polygonal number (except of course that each p is the second p-agonal number), many centered polygonal numbers are primes.
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